1. Field of the invention
The present invention relates to a method, an apparatus, and a program for restoring phase information to be used for constructing an image on the basis of image information obtained by radiation imaging. In this application, the word “radiation” is used in a broad sense that includes corpuscular beams such as an electron beam, and electromagnetic waves, in addition to X-rays, α-rays, β-rays, γ-rays, ultraviolet rays and the like.
2. Description of a Related Art
Conventionally, an imaging method using an X-ray, etc. is utilized in various fields, and particularly in the medical field, the method is one of the most important means for diagnosis. Since the first X-ray photograph was implemented, the X-ray photography method has been repeatedly improved, and a method using a combination of a fluorescent screen and an X-ray film is in the mainstream at present. On the other hand, in recent years, various digitized devices applying X-ray CT, ultrasonic waves, MRI, etc. are in practical use, and construction of a diagnostic information processing system or the like in hospitals is being promoted. With respect to X-ray images, many studies are also made for digitizing the imaging system. Digitizing the imaging system enables to store a large amount of data for a long period without degradation in image quality, and helps to make progress toward the medical diagnostic information processing system.
Now, a radiation image obtained as described above is generated by converting intensity of the radiation transmitted through an object into brightness of the image. For example, when performing imaging on a region including a bone part, the radiation transmitted through the bone part is largely attenuated, and the radiation transmitted through a region other than the bone part, i.e., a soft part is slightly attenuated. In this case, since the difference in intensity between the radiations transmitted through different tissues is large, a high-contrast radiation image can be obtained.
On the other hand, for example, when imaging a region of the soft part such as a breast, since radiation is easily transmitted through the soft part as a whole, the difference between tissues in the soft part is difficult to appear as the difference in intensity of the transmitted radiation. Therefore, only a low-contrast radiation image can be obtained with respect to the soft part. Thus, the conventional radiation imaging method is not suitable as a method of visualizing the slight difference between tissues in the soft part.
Here, as information included in the radiation transmitted through the object, there is phase information in addition to intensity information. Recently, a phase-contrast method of generating an image using this phase information is under study. The phase-contrast method is an image construction technology to convert phase difference produced by an X-ray, etc. transmitted through the object into brightness of an image.
The phase-contrast method includes a technique for obtaining phase difference on the basis of interference X-ray produced by using an interferometer or a zone plate, and a technique for obtaining phase difference on the basis of diffraction X-ray. Among these techniques, the technique for obtaining phase difference on the basis of diffraction X-ray (diffraction technique) is to obtain phase difference on the basis of the following principle. An X-ray, for example, propagates within a material because a wave progresses similarly to light. The propagation rate varies according to the refraction index of the material. Therefore, when applying an X-ray having uniform phase toward the object, the propagation rate of the X-ray varies according to the difference between tissues in the object. Thereby, the wavefront of the X-ray transmitted through the object is distorted and, as a result, diffraction fringes are produced in the X-ray image obtained on the basis of the transmitted X-ray. The pattern of the diffraction fringes differs in accordance with the distance between the screen on which the X-ray image is formed and the object, and the wavelength of the X-ray. Thus, analyzing two or more X-ray images having different patterns of diffraction fringes, phase differences of the X-ray generated in the respective positions on the screen can be obtained. By converting the phase differences into brightness, an X-ray image that clearly shows the difference between tissues in the object can be obtained.
Particularly, in the radiation after transmitted through a soft part of the object, since the difference in phase is larger than the difference in intensity in accordance with the difference between tissues through which the radiation has been transmitted, the subtle difference between tissues can be visualized by using the phase-contrast method. In order to use the above-described phase-contrast method, imaging conditions in radiation imaging or techniques for restoring phase information from patterns of diffraction fringes are under study.
B. E. Allman et al. “Noninterferometric quantitative phase imaging with soft x rays”, J. Optical Society of America A, Vol. 17, No. 10 (October 2000), pp. 1732-1743 discloses that an X-ray image is constructed by performing phase restoration on the basis of image information obtained by imaging with soft X-rays. In this reference, the basic equation of phase restoration, TIE (transport of intensity equation) is used.
                              κ          ⁢                                    ∂                              I                ⁡                                  (                  r                  )                                                                    ∂              z                                      =                              -                          ∇              ⊥                                ·                      {                                          I                ⁡                                  (                  r                  )                                            ⁢                                                ∇                  ⊥                                ⁢                                  ϕ                  ⁡                                      (                    r                    )                                                                        }                                              (        1        )            Where r=(x, y, z) is vector, and
      ∇    ⊥    ⁢      =          (                                    ∂                                                                      ∂            x                          ,                              ∂                                                                      ∂            y                              )      In addition, κ denotes wave number.
Next, a principle of the phase restoration will be described by referring to FIG. 21. As shown in FIG. 21, an X-ray having a wavelength λ is input from the left side of the drawing, transmitted through an object plane 1 and enters a screen 2 at a distance of z from the object plane 1. Here, assuming that the intensity and the phase of the X-ray in a position (x, y) on the screen 2 are I(x, y) and φ(x, y), respectively. In this case, relationship expressed by the following equation holds between the intensity I(x, y) and phase φ(x, y). Here, the intensity I is square of amplitude of wave.
                                                        2              ⁢                                                          ⁢              π                        λ                    ⁢                                    ∂                              I                ⁡                                  (                                      x                    ,                    y                                    )                                                                    ∂              z                                      =                              -            ∇                    ·                      {                                          I                ⁡                                  (                                      x                    ,                    y                                    )                                            ⁢                              ∇                                  ϕ                  ⁡                                      (                                          x                      ,                      y                                        )                                                                        }                                              (        2        )            Substituting κ=2π/λ into Eq. (2) and rewriting (x, y) components into vector r, TIE expressed by Eq. (1) is derived.
However, since the above TIE is difficult to be solved, TIE has been used mainly by performing approximation thereon. For example, T. E. Gureyev et al. “Hard X-ray quantitative non-interferometric phase-contrast imaging”, SPIE, Vol. 3659 (1999), pp. 356-364 discloses that an X-ray image is constructed by performing phase restoration on the basis of image information obtained by imaging with hard X-rays. In this reference, TIE expressed by Eq. (1) is approximated as follows. First, Eq. (1) is developed. In the following equations, the vector r in the above reference is rewritten into (x, y) components.
                                                                                          -                  κ                                ⁢                                                      ∂                                          I                      ⁡                                              (                                                  x                          ,                          y                                                )                                                                                                  ∂                    z                                                              =                            ⁢                                                (                                                                                    ∂                                                                                                                                              ∂                        x                                                              ,                                                                  ∂                                                                                                                                              ∂                        y                                                                              )                                ·                                  (                                                                                    I                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    ⁢                                                                        ∂                                                      ϕ                            ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                                                                ∂                          x                                                                                      ,                                                                  I                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    ⁢                                                                        ∂                                                      ϕ                            ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                                                                ∂                          y                                                                                                      )                                                                                                        =                            ⁢                                                                                          ∂                                                                                                                                  ∂                      x                                                        ⁢                                      (                                                                  I                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    ⁢                                                                        ∂                                                      ϕ                            ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                                                                ∂                          x                                                                                      )                                                  +                                                                            ∂                                                                                                                                  ∂                      y                                                        ⁢                                      (                                                                  I                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    ⁢                                                                        ∂                                                      ϕ                            ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                                                                ∂                          y                                                                                      )                                                                                                                          =                            ⁢                                                                    I                    ⁡                                          (                                              x                        ,                        y                                            )                                                        ⁢                                      (                                                                                                                        ∂                            2                                                    ⁢                                                      ϕ                            ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                                                                ∂                                                      x                            2                                                                                              +                                                                                                    ∂                            2                                                    ⁢                                                      ϕ                            ⁡                                                          (                                                              x                                ,                                y                                                            )                                                                                                                                ∂                                                      y                            2                                                                                                                )                                                  +                                                                                                      ⁢                                                                                          ∂                                              I                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                                                            ∂                      x                                                        ⁢                                                            ∂                                              ϕ                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                                                            ∂                      x                                                                      +                                                                            ∂                                              I                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                                                            ∂                      y                                                        ⁢                                                            ∂                                              ϕ                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                                                            ∂                      y                                                                                                                                              =                            ⁢                                                                    I                    ⁡                                          (                                              x                        ,                        y                                            )                                                        ⁢                                                            ∇                      2                                        ⁢                                          ϕ                      ⁡                                              (                                                  x                          ,                          y                                                )                                                                                            +                                                      ∇                                          I                      ⁡                                              (                                                  x                          ,                          y                                                )                                                                              ·                                      ∇                                          ϕ                      ⁡                                              (                                                  x                          ,                          y                                                )                                                                                                                                                    (        3        )            Where
      ∇    2    ⁢      =                            ∂          2                          ∂                      x            2                              +                        ∂          2                          ∂                      y            2                              
Approximating the second term on the right side of Eq. (3) to zero, the approximation equation expressed by the following equation (4) is obtained.
                                          ∂                          I              ⁡                              (                                  x                  ,                  y                                )                                                          ∂            z                          ≅                              -                                          I                ⁡                                  (                                      x                    ,                    y                                    )                                            κ                                ⁢                                    ∇              2                        ⁢                          ϕ              ⁡                              (                                  x                  ,                  y                                )                                                                        (        4        )            In Eq. (4), φ(x, y) can be obtained from I(x, y) by a solution method such as the finite element method.
In addition, T. E. Gureyev et al. “Quantitative In-Line Phase-Contrast Imaging with Multienergy X Rays”, Physical Review Letter, Vol. 86, No. 25 (2001), PP. 5827-5830 discloses that an X-ray imaging is performed by using three kinds of X-rays having different wavelengths and the phase restoration is performed on the basis of the obtained image information. In this reference, attention is given to the relationship between phase and intensity of the X-ray just after transmitted through the object and intensity of the X-ray in a position at a predetermined distance from the object. When performing the X-ray imaging, the structure shown in FIG. 22 is assumed. That is, as shown in FIG. 22, the three kinds of X-rays having wavelengths λ0, λ1, and λ2, respectively, are transmitted through the object 3, and enter the screen 2 that is disposed in a position at a distance R from the object plane 1.
In this case, assuming that r⊥=(x, y), the following relationship is held between the intensity I(r⊥, 0, λ0) and the phase φ(r⊥, 0, λ0) of the X-ray of wavelength λ0, which is just after transmitted through the object 3, and the intensity I (r⊥, R, λm) of the diffraction X-ray of wavelength λm which is detected on the screen 2. In the following Equation (5), I(r⊥, 0, λ0)=exp{−M(r⊥, 0, λ0)}.
                              A          ⁡                      (                                                                                M                    ⁡                                          (                                                                        r                          ⊥                                                ,                        0                        ,                                                  λ                          0                                                                    )                                                                                                                                        -                                                                  ∇                        2                                            ⁢                                              ϕ                        ⁡                                                  (                                                                                    r                              ⊥                                                        ,                            0                            ,                                                          λ                              0                                                                                )                                                                                                                                                                                                            ∇                      M                                        ·                                          ∇                                              ϕ                        ⁡                                                  (                                                                                    r                              ⊥                                                        ,                            0                            ,                                                          λ                              0                                                                                )                                                                                                                                          )                          =                  (                                                                      g                  0                                                                                                      g                  1                                                                                                      g                  2                                                              )                                    (        5        )            Where,
                    A        =                  (                                                                      -                  1                                                                              γ                  0                                                                              γ                  0                                                                                                      -                                      σ                    1                    3                                                                                                                    σ                    1                                    ⁢                                      γ                    1                                                                                                                    σ                    1                    4                                    ⁢                                      λ                    1                                                                                                                        -                                      σ                    2                    3                                                                                                                    σ                    2                                    ⁢                                      γ                    2                                                                                                                    σ                    2                    4                                    ⁢                                      λ                    2                                                                                )                                                              σ            m                    =                                    λ              m                                      λ              0                                      ,                              γ            m                    =                                    R              ⁢                                                          ⁢                              λ                m                                                    2              ⁢                                                          ⁢              π                                      ,                              g            m                    =                      ln            ⁢                          {                              I                ⁡                                  (                                                            r                      ⊥                                        ,                    R                    ,                                          λ                      m                                                        )                                            }                        ⁢                                                  ⁢                          (                                                m                  =                  0                                ,                1                ,                2                            )                                          
In Eq. (5), if ∇M·∇φ(r⊥, R, λ0) is sufficiently small, approximation can be as follows.
                                          (                                                                                -                    1                                                                                        γ                    0                                                                                                                    -                                          σ                      1                      3                                                                                                                                  σ                      1                                        ⁢                                          γ                      1                                                                                            )                    ⁢                      (                                                                                M                    ⁡                                          (                                                                        r                          ⊥                                                ,                        0                        ,                                                  λ                          0                                                                    )                                                                                                                                        -                                                                  ∇                        2                                            ⁢                                              ϕ                        ⁡                                                  (                                                                                    r                              ⊥                                                        ,                            0                            ,                                                          λ                              0                                                                                )                                                                                                                                          )                          =                  (                                                                      g                  0                                                                                                      g                  1                                                              )                                    (        6        )            
Further, from Eq. (6), the intensity and the phase of the X-ray just after transmitted through the object 3 are expressed as follows.
                              M          ⁡                      (                                          r                ⊥                            ,              0              ,                              λ                0                                      )                          =                              λ                          Δ              ⁢                                                          ⁢              λ                                ⁢                      (                                          g                0                            -                                                σ                                      -                    2                                                  ⁢                                  g                  1                                                      )                                              (        7        )                                          -                                    ∇              2                        ⁢                          ϕ              ⁡                              (                                                      r                    ⊥                                    ,                  0                  ,                                      λ                    0                                                  )                                                    =                                            2              ⁢                                                          ⁢              π                                      R              ⁢                                                          ⁢              Δ              ⁢                                                          ⁢              λ                                ⁢                      (                                          σ                ⁢                                                                  ⁢                                  g                  0                                            -                                                σ                                      -                    2                                                  ⁢                                  g                  1                                                      )                                              (        8        )            Where, Δλ=λ1−λ0 and σ≡σ1=λ1/λ0.
The phase φ(r⊥, R, λ0) can be obtained by performing inverse Laplacian computation on the Laplacian ∇2φ(r⊥, R, λ0) of the phase in Eq. (8). Further, a visible image representing the object can be obtained by converting this phase into brightness in the image. Thus, by using Eq. (8), the computation for phase restoration can be easily performed on the basis of a small number of irradiation images obtained by changing the wavelengths. Therefore, in T. E. Gureyev et al. “Quantitative In-Line Phase-Contrast Imaging with Multienergy X Rays”, Physical Review Letter, Vol. 86, No. 25 (2001), pp. 5827-5830, X-ray imaging is performed by using X-rays having three kinds of wavelengths (energy) λ0=3.8 Å (E0=3.3 keV), λ1=7.3 Å (E1=1.7 keV), and λ2=2.5 Å (E2=5.0 keV).
Although a thin object can be imaged at these wavelengths, there is a problem that the imaging method is unsuitable in the case where an object having a larger thickness such as a breast or a chest of a human body is to be imaged because X-ray absorption when transmitted through the object is too large at these wavelengths.
Further, in order to perform higher precision phase restoration, it is desired to use a high-definition (high resolution) screen constituted by a many number of detecting elements the size of which is made as small as possible.
However, in the case where the high-definition screen is used, there is a problem that the X-ray imaging becomes vulnerable to the influence of noise. In order to reduce the influence of noise, it is conceivable that the irradiation amount of X-ray is increased. However, in the case of a living organism such as a human body, there is a problem that the irradiation amount can not be increased for fear of exposure. By the way, in the above-described papers of B. E. Allman et al. and T. E. Gureyev et al. in which a non-living organism is imaged, the relationship between noise and irradiation amount is not described.